| L(s) = 1 | − 3-s − 5·7-s − 2·11-s − 2·13-s − 3·17-s + 4·19-s + 5·21-s − 3·23-s − 2·27-s + 9·29-s − 8·31-s + 2·33-s + 4·37-s + 2·39-s − 12·41-s − 20·43-s − 12·47-s + 10·49-s + 3·51-s + 9·53-s − 4·57-s − 12·59-s + 13·61-s − 8·67-s + 3·69-s + 6·71-s − 11·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s − 0.603·11-s − 0.554·13-s − 0.727·17-s + 0.917·19-s + 1.09·21-s − 0.625·23-s − 0.384·27-s + 1.67·29-s − 1.43·31-s + 0.348·33-s + 0.657·37-s + 0.320·39-s − 1.87·41-s − 3.04·43-s − 1.75·47-s + 10/7·49-s + 0.420·51-s + 1.23·53-s − 0.529·57-s − 1.56·59-s + 1.66·61-s − 0.977·67-s + 0.361·69-s + 0.712·71-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 79 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 133 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 13 T + 159 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 11 T + 171 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T - 99 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 175 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17 T + 261 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.865423935971807762624332632134, −9.506475633829259115093445921717, −8.648941851317676998996647537364, −8.629832395937400056432430558503, −7.997520529206816269097327056233, −7.50077225273052291730772601934, −6.85894899629911088195646300130, −6.76974666974495771617996427917, −6.38740297205727405645125170880, −5.86083429752214970460079173621, −5.30895564777781553229651654351, −5.06403487682614199907566503390, −4.46863615184067325381056622000, −3.78437682720816143306271942034, −3.16801221234984872867302573743, −3.06837399513500100572665776072, −2.23557546262456169949715625068, −1.46176869012503653864248898767, 0, 0,
1.46176869012503653864248898767, 2.23557546262456169949715625068, 3.06837399513500100572665776072, 3.16801221234984872867302573743, 3.78437682720816143306271942034, 4.46863615184067325381056622000, 5.06403487682614199907566503390, 5.30895564777781553229651654351, 5.86083429752214970460079173621, 6.38740297205727405645125170880, 6.76974666974495771617996427917, 6.85894899629911088195646300130, 7.50077225273052291730772601934, 7.997520529206816269097327056233, 8.629832395937400056432430558503, 8.648941851317676998996647537364, 9.506475633829259115093445921717, 9.865423935971807762624332632134
